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Constructive logic

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Constructive logic is a family of logics where proofs must be constructive (i.e., proving something means one must build or exhibit it, not just argue it “must exist” abstractly). No “non-constructive” proofs are allowed (like the classic proof by contradiction without a witness).

The main constructive logics are the following:

1. Intuitionistic logic

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Founder: L. E. J. Brouwer (1908, philosophy)[1][2] formalized by A. Heyting (1930)[3] and A. N. Kolmogorov (1932)[4]

Key Idea: Truth = having a proof. One cannot assert “ or not ” unless one can prove or prove .

Features:

  • No law of excluded middle ( is not generally valid).
  • No double negation elimination ( is not valid generally).
  • Implication is constructive: a proof of is a method turning any proof of P into a proof of Q.

Used in: Type theory, constructive mathematics.

2. Modal logics for constructive reasoning

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Founder(s):

Interpretation (Gödel): means “ is provable” (or “necessarily ” in the proof sense).

Further: Modern provability logics build on this.

3. Minimal logic

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Simpler than intuitionistic logic.

Founder: I. Johansson (1937)[6]

Key Idea: Like intuitionistic logic but without assuming the principle of explosion (ex falso quodlibet, “from falsehood, anything follows”).

Features:

  • Doesn’t automatically infer any proposition from a contradiction.

Used for: Studying logics without commitment to contradictions blowing up the system.

4. Intuitionistic type theory (Martin-Löf type theory)

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Founder: P. E. R. Martin-Löf (1970s)

Key Idea: Types = propositions, terms = proofs (this is the Curry–Howard correspondence).

Features:

  • Every proof is a program (and vice versa).
  • Very strict — everything must be directly constructible.

Used in: Proof assistants like Coq, Agda.

5. Linear logic

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Not strictly intuitionistic, but very constructive.

Founder: J. Girard (1987)[7]

Key Idea: Resource sensitivity — one can only use an assumption once unless one specifically says it can be reused.

Features:

  • Tracks “how many times” you can use a proof.
  • Splits conjunction/disjunction into multiple types (e.g., additive vs. multiplicative).

Used in: Computer science, concurrency, quantum logic.

6. Other Constructive Systems

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  • Topos Logic: Internal logics of topoi (generalized spaces) are intuitionistic.

See also

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Notes

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References

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  • Berka, Karel; Kreiser, Lothar, eds. (1986). Logik-Texte. De Gruyter. doi:10.1515/9783112645826.
  • Kolmogorov, Andrey (1932). "On the Principle of Excluded Middle". Mathematical Logic Quarterly. 10: 65–74.